Q-fuzzy structure on JU-algebra

Definition 1

⁷ JU-algebra is an algebra $(X,◦,1)$ of type $(2,0)$ with a binary operation $◦$ and a fixed element 1 if it holds , $\forall x,y,z\in X$ :

In a JU-algebra X, we can define a partial ordering “≤” by Putting $x\le y$ if and only $y◦x=1$ .

Lemma 1

^7,9 If X is a JU-algebra, then the following hold for any $x,y,z\in X$ :

Definition 2

⁸A nonempty subset S of a JU-algebra X is called a subalgebra of X if $x◦y\in S,\forall x,y\in S.$

Definition 3

⁸A nonempty subset J of a JU-algebra X is said to be a JU-ideal of X if it satisfies:

Definition 4

³A Q-fuzzy set in a nonempty set X (or a Q-fuzzy subset of X) is an arbitrary function $\eta :X\times Q\to [0,1],$ where Q is a nonempty set and [0, 1] is the unit segment of the real line.

Definition 5

³let η be a Q-fuzzy set in X, $\forall t\in$ $[0,1]$ .The set $U(\eta ;t,q)=\{x\in X|\eta \left(x\right)\ge t\}$ is called an upper-level subset of η. And the set $L(\eta ;t,q)=\{x\in X|\eta \left(x\right)\le t\}$ is called a lower-level subset of η.

Definition 6

¹⁰If η be a Q-fuzzy set in a set X. Then the complement denoted by $\mathit{\eta c}$ , is the Q-fuzzy subset of X given by $\mathit{\eta c}(x,q)=1-\eta (x,q),\forall x\in X$ and $q\in Q.$

Definition 7

⁴let $\mu$ be a fuzzy subset of a JU-algebra X and $\alpha \in [0,1].$ The fuzzy sets ${\mu }^{\alpha }$ and ${\mu }_{\alpha }$ of X are defined as follows:

${\mu }^{\alpha }=\mathit{min}\{\mu \left(x\right),\alpha \}$ and ${\mu }_{\alpha }=\mathit{max}\{\mu \left(x\right),1-\alpha \},\forall x\in X.$ It is evident that ${\mu }^1=\mu$ , ${\mu }^0=0$ , and ${\mu }_1=\mu ,{\mu }_0=\mu$

Definition 8

⁶A fuzzy set μ of a JU-algebra X is said to be normal if there exists $x\in X$ such that $\mu \left(x\right)=1$ .

Definition 9

Let X be a JU-algebra, a Q-fuzzy set η in X is called Q-fuzzy JU-subalgebra of X if for any $x,y\in X$

Example 1

Let $X=\{1,2,3,4\}$ in which $◦$ is defined by the following table ( Table 1).⁷ Clearly, $(X,◦,1)$ is a JU-algebra. Let $Q=\{q_1,q_2,q_3\}$ and we define a Q-fuzzy set $\eta :Q\times X\to [0,1]$ : by Routine calculation gives that $\eta$ is a Q-fuzzy JU-subalgebra of X ( Table 2).

Lemma 2

If η is a Q-fuzzy JU-subalgebra of a JU-algebra X. Then $\eta (1,q)\ge \eta (x,q),\forall x\in X$ and $\forall q\in Q$ .

Proof:

Suppose η be a Q-fuzzy JU-subalgebra of a JU-algebra X.

Since, $\eta (1,q)=\eta (x◦x,q)$ ............... (Lemma no. 1 c)

      $\ge \mathit{min}\{\eta (x,q),\eta (x,q)\}$

      $=\eta (x,q)$

Definition 10

Let η be a Q-fuzzy set in a JU-algebra X, $\forall t\in [0,1].$ The set $\eta =\{x\in X|\eta (x,q)\ge t\}$ is called an upper-level subset of η. And the set ${\eta }_t=\{x\in X|\eta \left(x\right)\le t\}$ is called a lower-level subset of η.

Clearly_, $\mathit{\eta t}\cup \mathit{\eta t}=X$ for any $t\in [0,1]$ if $t_1\le t_2$ , then ${\eta }_{t_1}\subseteq {\eta }_{t_2}$ but if $t_2\le t_1,$ then ${\eta }_{t_2}\subseteq {\eta }_{t_1}$ _.

Theorem 3

A Q-fuzzy set η of a JU-algebra X is a Q-fuzzy subalgebra if and only if for every $t\in [0,1],q\in Q$ , and $\mathit{\eta t}$ is a JU-subalgebra of X.

Proof: Suppose that $\eta$ is a Q-fuzzy subalgebra of a JU-algebra X and ${\eta }_t\ne \varnothing$ . For any $x,y\in {\eta }_t$ such that $\eta (x,q)\le t$ and $\eta (y,q)\le t$ implies that

Hence, ${\eta }_t$ is a subalgebra of X.

Conversely, if η_t is a subalgebra of X. Let x, y $\in$ η_t and there exist $t_1,t_2\in [0,1]$ such that $\eta (x,q)\le t_1$ and $\eta (y,q)\le t_2$ .

Take, $=\mathit{min}\{t_1,t_2\}$ . By assumption ${\eta }_t$ is a subalgebra of X, we have $x◦y\in {\eta }_t$ then $\eta (x◦y)\le t=\mathit{min}\{\eta (x,q),\eta (y,q)\}\eta$ is a Q-fuzzy JU-subalgebra of X.

Theorem 4

A Q-fuzzy set η of a JU-algebra X is a Q-fuzzy subalgebra if and only if for every $t\in [0,1]$ and $q\in Q$ , $\mathit{\eta t}$ is a JU-subalgebra of X.

Proof:

It is straight forward by theorem 3.

Definition 11

A Q-fuzzy set η of a JU-algebra X is called a Doubt Q-fuzzy JU-subalgebra of X if $\forall x,y\in X$ and $q\in Q$

Example 2

Let $X=\{1,2,3,4\}$ in which $◦$ is defined by the following table ( Table 3).⁷

Clearly, $(X,◦,1)$ is a JU-algebra. Let $Q=\{q_1,q_2\}$ and we define a Q-fuzzy set $\eta :Q\times X\to [0,1]$ by:

Routine calculation gives that η is a Doubt Q-fuzzy JU-subalgebra of X ( Table 4).

Lemma 5

If $\eta$ is a Doubt Q-fuzzy JU-subalgebra of a JU-algebra X. Then $\eta (1,q)\le \eta (x,q),\forall x\in X$ and $\forall q\in Q$ .

Proof:

Suppose η be a Doubt Q-fuzzy JU-subalgebra of a JU- algebra X.

Since, $(1,q)=\eta (x◦x,q)$ .................. (Lemma no. 1 c)

      $\ge \mathit{max}\{\eta (x,q),\eta (x,q)\}$

      $=\eta (x,q)$

Theorem 6

For every subalgebra of a JU-algebra X, it can be realized as a level subalgebra of some Doubt Q-fuzzy JU-subalgebra of X.

Proof:

Let X be a subalgebra of a JU-algebra X, and let η be a Q-fuzzy set in X defined by:

Where $t\in [0,1]$ is fixed. Clearly, $\mathit{\eta t}\cup \mathit{\eta t}=X$ .

We went to prove that η is a Doubt Q-fuzzy subalgebra of X.

If $x,y\in X$ and $q\in Q$ then $x◦y\in X$

If $x,y\notin X$ and $q\in Q$ then $\eta (x,q)=\eta (y,q)=0$

$\eta (x◦y,q)\le \mathit{max}\{\eta (x,q),\eta (y,q)\}=0$

If at most one of $x,y\in X$ and $q\in Q$ then at least one of $\eta (x,q)$ and $\eta (y,q)$ is equal to t. Thus

Theorem 7

A Q-fuzzy subset η of a JU-algebra X is a Q-fuzzy subalgebra of X if and only if its complement η^c is a Doubt Q-fuzzy subalgebra of X.

Proof:

Let η be a Q-fuzzy subalgebra of X and let x, y $\in$ X.

Then

Conversely, suppose is a Doubt Q-fuzzy subalgebra of X.

Definition 12

A Q-fuzzy JU-subalgebra η of X is said to be normal if there exists $x\in X$ such that $\eta (x,q)=1$

Example 3

Consider $X=\{1,2,3,4,5\},$ we construct the following table.

Clearly, (X, ◦, 1) is a JU-algebra ( Table 5).⁷

Let $Q=\{q_1,q_2\},$ given a Q-fuzzy set $\eta :Q\times X\to [0,1]$ then it gives η is a normal Q-fuzzy JU-subalgebra of X( Table 6).

Lemma 8

If η is a normal Q-fuzzy JU-subalgebra of a JU-algebra X, then $\eta (1,q)=1$

Proof:

Suppose η be normal Q-fuzzy JU-subalgebra of a JU-algebra X.

Since, $\eta (1,q)\ge \eta (x,q)=1$ .......................( Lemma 2)

$\Rightarrow \eta (1,q)\ge 1$ But, $\eta (1,q)\le 1$

Therefore, $\eta (1,q)=1$ .

Theorem 9

For a Q-fuzzy JU-subalgebra of η of X, we can generate a normal fuzzy subalgebra of X which contains η.

Proof: Let η be a Q-fuzzy JU-subalgebra of X. Define a Q-fuzzy subset ${\eta }^{\ast }$ of X as:

Let $x,y\in X$ , ${\eta }^{\ast }(x◦y,q)=\eta (x◦y,q)+{\eta }^c(1,q)$

And

$\Rightarrow$ is normal Q-fuzzy JU-algebra of X. Clearly, $\eta \subseteq {\eta }^{\ast }$ thus ${\eta }^{\ast }$ is a normal JU-subalgebra of X which contains η.

Corollary 10

If η itself is normal then $\eta ={\eta }^{\ast }$ If η is a Q-fuzzy JU-subalgebra of X then $\left({\eta }^{\ast }\right)\ast ={\eta }^{\ast }$

Proof:

It is straight forward by theorem 9.

Theorem 11

Let η be a Q-fuzzy JU-subalgebra of X. If η contains a normal Q-fuzzy JU-subalgebra of X generated by any other Q-fuzzy JU-subalgebra of X then η is normal.

Proof:

Let λ be a Q-fuzzy JU-subalgebra of X. by theorem 9 and ${\lambda }^{\ast }$ is a normal Q-fuzzy JU-subalgebra of X. i.e. ${\lambda }^{\ast }=1$ by Lemma 8. Then let η is a Q-fuzzy JU-subalgebra of X such that ${\lambda }^{\ast }\subseteq \eta$ then

Put x = 1 $\Rightarrow$ η (1, q) ≥ ${\lambda }^{\ast }$ = 1

$\Rightarrow \eta (1,q)\ge 1$ . Hence η is normal.

Theorem 12

Let η be a Q-fuzzy JU-subalgebra of X and let $f:[1,\eta (1,q)]\to f[0,1]$ be an increasing function. Define a Q-fuzzy set $\eta :X\to [0,1]$ by $\eta =f\left(\eta (x,q)\right),\forall x\in X$ then

Proof:

Theorem 13

Let η be a Doubt Q-fuzzy JU-subalgebra of X and let $g:[1,\eta (1,q)]\to [0,1]$ be a decreasing function. Define a Q-fuzzy set $\eta :X\to [0,1]$ by ${\eta }^g=g\left(\eta (x,q)\right),\forall x\in X$ then

Proof:

Definition 13

Let X be a JU-algebra, a Q-fuzzy set η in X is called Q-fuzzy JU-ideal of X if it satisfies the following conditions:

Example 4

(See table 3.1 from example 3) consider $X=\{1,2,3,4\}$ , and let $Q=\{q_1,q_2\},$ given a Q-fuzzy set $\eta :Q\times X\to [0,1]$ by Routine calculation gives that η is a Q-fuzzy JU-ideal of X ( Table 7).

Theorem 14

Every Q-fuzzy JU-ideals of a JU-algebra X is order reversing.

Proof:

Let $\eta$ be a Q-fuzzy JU-ideals of a JU-algebra X and for any $x,y\in X$ be such that $x\le y$ , then $y◦x=1$

Theorem 15

If η is a Q-fuzzy JU-ideals of X, then η ((x ◦ y) ◦ y, q) ≥ η(x, q).

Proof:

Since (x◦ [(x◦y) ◦y]) = 1.......................... (Lemma no.1e)

Now, $\eta ((x◦y)◦y,q)\ge \mathit{min}\{\eta (x,q),\eta (x◦[(x◦y)◦y])\}$ , $\forall q\in Q=\mathit{min}\{\eta (x,q),\eta (1,q)\}=\eta (x,q)$

Theorem 16

A Q-fuzzy ideal of a JU- algebra X is Q-fuzzy JU-subalgebra when $\eta (x◦(x◦y),q)\ge \eta (y,q),$ for any $x,y\in X$ and $\forall q\in Q$ .

Proof:

Suppose $\eta$ is a Q-fuzzy ideal of a JU- algebra X and $\eta (x◦(x◦y),q)\ge \eta (y,q).$ Then for any $x,y\in X$ and $\forall q\in Q$ .

Theorem 17

Let η be a Q-fuzzy set in a JU-algebra X and let $t\in [0,1].$ Then η is a Q-fuzzy JU-ideal of X if and only if the upper level subset ${\eta }^t$ is a JU-ideal of X.

Proof:

Suppose η be a Q-fuzzy JU-ideal in a JU-algebra X and ${\eta }^t\ne \varnothing$

Let $x,x◦y\in {\eta }_t$ and $\forall q\in Q$ implies that $\eta (x,q)\ge t$ and $\eta (x◦y,q)\ge t$ then $\eta (y,q)\ge \mathit{min}\{\eta (x,q),\eta (x◦y,q)\}\ge \mathit{min}\{t,t\}=t$

This implies that y $\in {\eta }^t$ and hence ${\eta }^t$ is a JU-ideal of X.

Conversely, assume that the upper level subset η^t is a JU-ideal of X.

Further, let we assume that there exist some x₀ $\in$ X such that $\eta (1,q)<\eta (x_0,q)$

Take $s=\frac{1}{2}[\eta (1,q)\ast \eta (x_0,q)]$

$x_0\in {\eta }_s$ and $1\notin {\eta }_s$ , which is contradict to our assumption that ${\eta }_s$ is a JU-ideal of X.

Therefore, $\eta (1,q)\ge \eta (x,q),\forall x\in X$ .

And

assume that $x_0,y_0\in X$ such that

Take $s=\frac{1}{2}[\eta (y_0,q)+\mathit{min}\{\eta (x_0,q),\eta (x_0◦y_0,q)\}]$

$y_0\notin {\eta }_s$ , a contradiction, since ${\eta }_s$ is a JU-ideal of X. Therefore, $\eta (y_0,q)\ge \mathit{min}\{\eta (x_0,q),\eta (x_0◦y_0),q\}$ for any $x,y\in X$ and $\forall q\in Q.$

Theorem 22

For a Q-fuzzy JU-ideal of η of X, we can generate a normal fuzzy ideal of X which contains η.

Proof:

Let η be a Q-fuzzy JU-ideal of X. Define a Q-fuzzy subset ${\eta }^{\ast }$ of X as:

Let $x,y\in X,\eta \ast (1,q)=\eta (1,q)+{\eta }_c(1,q)$

$\ge \eta (x,q)+{\eta }_c(1,q)={\eta }^{\ast }(x,q)$ And

$\Rightarrow {\eta }^{\ast }$ is normal Q-fuzzy JU-algebra of X.

Clearly $\eta \subseteq {\eta }^{\ast }$ thus ${\eta }^{\ast }$ is a normal JU-ideal of X which contains η.

Corollary 23

If η itself is normal then $\eta ={\eta }^{\ast }$ . And if $\eta$ is a Q-fuzzy JU-ideal of X then $\left({\eta }^{\ast }\right)\ast ={\eta }^{\ast }$

Proof:

It is straight forward by theorem 22.

Theorem 24

Let η is a Q-fuzzy JU-ideal of X. If η contains a normal Q-fuzzy JU-ideal of X generated by any other Q-fuzzy JU- ideal of X then η is normal.

Proof:

Let λ be a Q-fuzzy JU- ideal of X. By theorem 22 $\lambda \ast$ is a normal Q-fuzzy JU-ideal of X. i.e., $\lambda \ast$ = 1 by Lemma 21

Let η is a Q-fuzzy JU-ideal of X such that $\mathrm{\lambda }\ast \subseteq \mathrm{\eta }$ then $\eta (x,q)\ge {\lambda }^{\ast }(x,q),\forall x\in X$ Put $x=1$

$\Rightarrow \eta (1,q)\ge 1.$ but, $\eta (1,q)\le 1$

Therefore, $\eta (1,q)=1.$

Theorem 25

Let η be a Q-fuzzy JU- ideal of X and let $f:[1,\eta (1,q)]\to [0,1]$ be an increasing function. Define a Q-f fuzzy set ${\eta }_f:X\to [0,1]$ by $\eta =f\left(\eta (x,q)\right),\forall x\in X$ then

Proof:

Theorem 26

Let η be a Doubt Q-fuzzy JU- ideal of X and let $g:[1,\eta (1,q)]g\to [0,1]$ be a decreasing function. Define $g$ a Q-fuzzy set $\eta :X\to [0,1]\mathit{by}\eta =g\left(\eta (x,q)\right),\forall x\in X$ then

Proof: